About Convergence of complex sequence
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From: Eric on 16 Apr 2010 21:25 On 4æ17æ¥, ä¸å2æ19å, "Dave L. Renfro" <renfr...(a)cmich.edu> wrote: > Robert Israel wrote: > >> Hint: how can |Z_n - a| < E and |Z_n - b| < E produce a contradiction? > Eric wrote: > > Thank you very much. > > Maybe we should suppose it has two limit point.And find the > > contradiction.Sorry I have no idea. > > |Z_n - a| < E is the interior of the circle with center a > and radius sqrt(E). |Z_n - b| < E is the interior of the > circle with center b and radius sqrt(E). Draw a diagram > of the two circular regions when E is very small . . . > > Dave L. Renfro Thank you. It's means that Z_n -> a as sqrt(E)-> 0 and Z_n -> b as sqrt(E)-> 0 so a=b (It's contradiction) Thank you very much Eric.
From: David C. Ullrich on 17 Apr 2010 04:41 On Fri, 16 Apr 2010 10:53:04 -0700 (PDT), Eric <eric955308(a)yahoo.com.tw> wrote: >On 4��17��, �W��1��37��, Robert Israel ><isr...(a)math.MyUniversitysInitials.ca> wrote: >> Eric <eric955...(a)yahoo.com.tw> writes: >> > On 4=E6=9C=8816=E6=97=A5, =E4=B8=8B=E5=8D=886=E6=99=8215=E5=88=86, William >> > = >> > Elliot <ma...(a)rdrop.remove.com> wrote: >> > > On Fri, 16 Apr 2010, David C. Ullrich wrote: >> > > > <eric955...(a)yahoo.com.tw> wrote: >> >> > > >> How to prove the sequence just has one limit point (Proof by >> > > >> contradiction ). >> >> > > > _What_ sequence? Some sequences have more than one >> > > > limit point. >> >> > > > A given sequence has at most one limit (some sequences do not >> > > > have a limit, because they have more than one limit point...) >> >> > > By limit point, do you mean a cluster point? >> >> > > > What did the problem actually ask? >> >> > > What are the actual definitions being used? >> >> > Tank you for your interest.My teacher talk about the "Conergence of >> > complex sequences" >> > Def.{Z_n} is convergent to the limit a iff >> > for all E>0, there exist N>0:for all n>N =3D>|Z_n-a|<E >> > this is almost mean that Z_n -> a as n -> infinite >> > and he introduce (Cauchy criterion) and use it to prove that >> > definition (Def) >> >> > and then he gave us this problem (How to prove that has one limit >> > point (Proof by contradiction ). >> >> I think there is a language barrier here. "Limit point" is not the same as >> "limit". The actual problem must have been to prove that a sequence has at >> most one limit. >> >> OK, let's get you started with the proof by contradiction. >> >> Hint: how can |Z_n - a| < E and |Z_n - b| < E produce a contradiction? >> -- >> Robert Israel isr...(a)math.MyUniversitysInitials.ca >> Department of Mathematics http://www.math.ubc.ca/~israel >> University of British Columbia Vancouver, BC, Canada- ���óQ�ޥΤ�r - >> >> - ��ܳQ�ޥΤ�r - > >Thank you very much. >Maybe we should suppose it has two limit point.And find the >contradiction.Sorry I have no idea. When you post questions about homework you should at least _read_ the replies! The problem was (almost certainly) to show that a sequence has at most one _limit_, not "limit point".
From: Tonico on 17 Apr 2010 04:54 On Apr 17, 11:41 am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > On Fri, 16 Apr 2010 10:53:04 -0700 (PDT), Eric > > > > > > <eric955...(a)yahoo.com.tw> wrote: > >On 4¤ë17¤é, ¤W¤È1®É37¤À, Robert Israel > ><isr...(a)math.MyUniversitysInitials.ca> wrote: > >> Eric <eric955...(a)yahoo.com.tw> writes: > >> > On 4=E6=9C=8816=E6=97=A5, =E4=B8=8B=E5=8D=886=E6=99=8215=E5=88=86, William > >> > = > >> > Elliot <ma...(a)rdrop.remove.com> wrote: > >> > > On Fri, 16 Apr 2010, David C. Ullrich wrote: > >> > > > <eric955...(a)yahoo.com.tw> wrote: > > >> > > >> How to prove the sequence just has one limit point (Proof by > >> > > >> contradiction ). > > >> > > > _What_ sequence? Some sequences have more than one > >> > > > limit point. > > >> > > > A given sequence has at most one limit (some sequences do not > >> > > > have a limit, because they have more than one limit point...) > > >> > > By limit point, do you mean a cluster point? > > >> > > > What did the problem actually ask? > > >> > > What are the actual definitions being used? > > >> > Tank you for your interest.My teacher talk about the "Conergence of > >> > complex sequences" > >> > Def.{Z_n} is convergent to the limit a iff > >> > for all E>0, there exist N>0:for all n>N =3D>|Z_n-a|<E > >> > this is almost mean that Z_n -> a as n -> infinite > >> > and he introduce (Cauchy criterion) and use it to prove that > >> > definition (Def) > > >> > and then he gave us this problem (How to prove that has one limit > >> > point (Proof by contradiction ). > > >> I think there is a language barrier here. "Limit point" is not the same as > >> "limit". The actual problem must have been to prove that a sequence has at > >> most one limit. > > >> OK, let's get you started with the proof by contradiction. > > >> Hint: how can |Z_n - a| < E and |Z_n - b| < E produce a contradiction? > >> -- > >> Robert Israel isr...(a)math.MyUniversitysInitials.ca > >> Department of Mathematics http://www.math.ubc.ca/~israel > >> University of British Columbia Vancouver, BC, Canada- ÁôÂóQ¤Þ¥Î¤å¦r - > > >> - Åã¥Ü³Q¤Þ¥Î¤å¦r - > > >Thank you very much. > >Maybe we should suppose it has two limit point.And find the > >contradiction.Sorry I have no idea. > > When you post questions about homework you should at least > _read_ the replies! The problem was (almost certainly) to show that > a sequence has at most one _limit_, not "limit point".- Perhaps so and perhaps, as somebody else alread pointed out, it is a language barrier. I think your first response was likely to cause confusion, as it seemingly did, in the OP due to his/her obvious beginner level. It was likely to think that he/she would confuse or wouldn't distinguish between "sequence having a limit" and "limit point(s) of a given sequence". So perhaps a more or less direct answer could clear doubts out: suppose {a_n} is a seq. that has two different limits A,B, say |A - B| = K, for some some constant K > 0 ==> taking e = K/2 > 0 we know there exist natural numbers N1, N2 s.t: 1) |a_n - A| < e for all n > N1 2) |a_n - B| < e for all n > N2 Let now M:= Max(N1,N2) (the greatest between these two numbers N1, N2) ==> for all n > M both equalities (1)-(2) are true , but then we get for all n > M: |A - B| = |A - a_n + a_n - B| <= |A - a_n| + |a_n - B| < 2e ...and now can you ( the OP) see the contradiction we get? Tonio
From: Eric on 17 Apr 2010 05:28 On 4æ17æ¥, ä¸å4æ41å, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > On Fri, 16 Apr 2010 10:53:04 -0700 (PDT), Eric > > > > > > <eric955...(a)yahoo.com.tw> wrote: > >On 4¤ë17¤é, ¤W¤Ã1®Ã37¤Ã, Robert Israel > ><isr...(a)math.MyUniversitysInitials.ca> wrote: > >> Eric <eric955...(a)yahoo.com.tw> writes: > >> > On 4=E6=9C=8816=E6=97=A5, =E4=B8=8B=E5=8D=886=E6=99=8215=E5=88=86, William > >> > = > >> > Elliot <ma...(a)rdrop.remove.com> wrote: > >> > > On Fri, 16 Apr 2010, David C. Ullrich wrote: > >> > > > <eric955...(a)yahoo.com.tw> wrote: > > >> > > >> How to prove the sequence just has one limit point (Proof by > >> > > >> contradiction ). > > >> > > > _What_ sequence? Some sequences have more than one > >> > > > limit point. > > >> > > > A given sequence has at most one limit (some sequences do not > >> > > > have a limit, because they have more than one limit point...) > > >> > > By limit point, do you mean a cluster point? > > >> > > > What did the problem actually ask? > > >> > > What are the actual definitions being used? > > >> >  Tank you for your interest.My teacher talk about the "Conergence of > >> > complex sequences" > >> > Def.{Z_n} is convergent to the limit a iff > >> > for all E>0, there exist N>0:for all n>N =3D>|Z_n-a|<E > >> > this is almost mean that Z_n -> a as n -> infinite > >> > and he introduce (Cauchy criterion) and use it to prove that > >> > definition (Def) > > >> > and then he gave us this problem (How to prove that has one limit > >> > point (Proof by contradiction ). > > >> I think there is a language barrier here.  "Limit point" is not the same as > >> "limit".  The actual problem must have been to prove that a sequence has at > >> most one limit.  > > >> OK, let's get you started with the proof by contradiction.  > > >> Hint: how can |Z_n - a| < E and |Z_n - b| < E produce a contradiction? > >> -- > >> Robert Israel        isr...(a)math.MyUniversitysInitials.ca > >> Department of Mathematics     http://www.math.ubc.ca/~israel > >> University of British Columbia       Vancouver, BC, Canada- ÃôÃóQ¤Ã¥äå¦r - > > >> - à ã¥Ã³Q¤Ã¥äå¦r - > > >Thank you very much. > >Maybe we should suppose it has two limit point.And find the > >contradiction.Sorry I have no idea. > > When you post questions about homework you should at least > _read_ the replies! The problem was (almost certainly) to show that > a sequence has at most one _limit_, not "limit point".- é±è被å¼ç¨æå - > > - 顯示被å¼ç¨æå - Sorry,I'm not sure about it,so I can't replies this problem. Sorry but thank you for your help Thank you very much. Eric.
From: Eric on 17 Apr 2010 05:31
On 4æ17æ¥, ä¸å4æ54å, Tonico <Tonic...(a)yahoo.com> wrote: > On Apr 17, 11:41 am, David C. Ullrich <ullr...(a)math.okstate.edu> > wrote: > > > > > > > On Fri, 16 Apr 2010 10:53:04 -0700 (PDT), Eric > > > <eric955...(a)yahoo.com.tw> wrote: > > >On 4¤ë17¤é, ¤W¤Ã1®Ã37¤Ã, Robert Israel > > ><isr...(a)math.MyUniversitysInitials.ca> wrote: > > >> Eric <eric955...(a)yahoo.com.tw> writes: > > >> > On 4=E6=9C=8816=E6=97=A5, =E4=B8=8B=E5=8D=886=E6=99=8215=E5=88=86, William > > >> > = > > >> > Elliot <ma...(a)rdrop.remove.com> wrote: > > >> > > On Fri, 16 Apr 2010, David C. Ullrich wrote: > > >> > > > <eric955...(a)yahoo.com.tw> wrote: > > > >> > > >> How to prove the sequence just has one limit point (Proof by > > >> > > >> contradiction ). > > > >> > > > _What_ sequence? Some sequences have more than one > > >> > > > limit point. > > > >> > > > A given sequence has at most one limit (some sequences do not > > >> > > > have a limit, because they have more than one limit point...) > > > >> > > By limit point, do you mean a cluster point? > > > >> > > > What did the problem actually ask? > > > >> > > What are the actual definitions being used? > > > >> >  Tank you for your interest.My teacher talk about the "Conergence of > > >> > complex sequences" > > >> > Def.{Z_n} is convergent to the limit a iff > > >> > for all E>0, there exist N>0:for all n>N =3D>|Z_n-a|<E > > >> > this is almost mean that Z_n -> a as n -> infinite > > >> > and he introduce (Cauchy criterion) and use it to prove that > > >> > definition (Def) > > > >> > and then he gave us this problem (How to prove that has one limit > > >> > point (Proof by contradiction ). > > > >> I think there is a language barrier here.  "Limit point" is not the same as > > >> "limit".  The actual problem must have been to prove that a sequence has at > > >> most one limit.  > > > >> OK, let's get you started with the proof by contradiction.  > > > >> Hint: how can |Z_n - a| < E and |Z_n - b| < E produce a contradiction? > > >> -- > > >> Robert Israel        isr...(a)math.MyUniversitysInitials.ca > > >> Department of Mathematics     http://www.math..ubc.ca/~israel > > >> University of British Columbia       Vancouver, BC, Canada- ÃôÃóQ¤Ã¥äå¦r - > > > >> - à ã¥Ã³Q¤Ã¥äå¦r - > > > >Thank you very much. > > >Maybe we should suppose it has two limit point.And find the > > >contradiction.Sorry I have no idea. > > > When you post questions about homework you should at least > > _read_ the replies! The problem was (almost certainly) to show that > > a sequence has at most one _limit_, not "limit point".- > > Perhaps so and perhaps, as somebody else alread pointed out, it is a > language barrier. I think your first response was likely to cause > confusion, as it seemingly did, in the OP due to his/her obvious > beginner level. It was likely to think that he/she would confuse or > wouldn't distinguish between "sequence having a limit" and "limit > point(s) of a given sequence". > > So perhaps a more or less direct answer could clear doubts out: > suppose {a_n} is a seq. that has two different limits A,B, say |A - B| > = K, for some some constant K > 0 ==> taking > e = K/2 > 0 we know there exist natural numbers N1, N2 s.t: > > 1) |a_n - A| < e  for all n > N1 > > 2) |a_n - B| < e  for all n > N2 > > Let now  M:= Max(N1,N2) (the greatest between these two numbers N1, > N2) ==> for all n > M both equalities (1)-(2) are true , but then we > get for all n > M: > > |A - B| = |A - a_n + a_n - B| <= |A - a_n| + |a_n - B| < 2e ...and now > can you ( the OP) see the contradiction we get? > > Tonio- é±è被å¼ç¨æå - > > - 顯示被å¼ç¨æå - Thank you You help me to solve a very big problem. Thank you very much |